Question #43656

A beam of photons corresponding to the series limit of a hydrogen-like atom is used to illuminate a Young's Double Slit apparatus. The transition originates from a highly excited state with an effective lifetime $\tau$, decaying to a stable ground state. Due to the Heisenberg uncertainty principle ($\Delta E \Delta t \ge \frac{h}{4\pi}$, where $\Delta t = \tau$), the emitted light is not perfectly monochromatic but has a natural energy uncertainty $\Delta E$, corresponding to a wavelength spread $\Delta \lambda$. Assuming the fringe visibility drops to zero when the $m$-th order bright fringe of the central wavelength $\lambda$ coincides with the nearest dark fringe of wavelength $\lambda + \Delta \lambda$, what is the maximum observable order of interference $m$? (Assume $\Delta \lambda \ll \lambda$ and $c$ is the speed of light).